Question: Captain Emily has a ship, the H.M.S Crimson Lynx. The ship is four furlongs from the dread pirate Ashley and her merciless band of thieves. If her ship hasn't already been hit, Captain Emily has probability $\dfrac{1}{2}$ of hitting the pirate ship. If her ship has been hit, Captain Emily will always miss. If her ship hasn't already been hit, dread pirate Ashley has probability $\dfrac{1}{6}$ of hitting the Captain's ship. If her ship has been hit, dread pirate Ashley will always miss. If the Captain and the pirate each shoot once, and the Captain shoots first, what is the probability that the Captain hits the pirate ship, but the pirate misses?
Answer: The probability of event A happening, then event B, is the probability of event A happening times the probability of event B happening given that event A already happened. In this case, event A is the Captain hitting the pirate ship and event B is the pirate missing the Captain's ship. The Captain fires first, so her ship can't be sunk before she fires her cannons. So, the probability of the Captain hitting the pirate ship is $\dfrac{1}{2}$. If the Captain hit the pirate ship, the pirate has no chance of firing back. So, the probability of the pirate missing the Captain's ship given the Captain hitting the pirate ship is $1$. The probability that the Captain hits the pirate ship, but the pirate misses is then the probability of the Captain hitting the pirate ship times the probability of the pirate missing the Captain's ship given the Captain hitting the pirate ship. This is $\dfrac{1}{2} \cdot 1 = \dfrac{1}{2}$